Go to https://www.icloud.com/#settings
You can then select to restore Reminders. Restore back to the earliest date available and it will work again (Although you can no longer access them on iCloud via the web or any devices that have been upgraded to iOS 13/Catalina. Your restored Reminders will also no longer Sync with iOS 13/Catalina. See update below for workaround).
This will also cause Calendars to be restored, so before restoring, back up your calendars. You can also install a Google account on your Mac (Go to System Preferences > Accounts and enter your Google details) and copy your calendar events to the Google Account, which will keep them backed up.
If you want to continue to use your Reminders and sync them between devices while still remaining on Mojave (or pre IOS 13), consider adding one of the following accounts to your devices. All these allow Reminders to continue to work after the upgrade even on Apple systems that cannot be upgraded to the new Reminders:
- Yahoo Account
- AOL Account
- Microsoft Exchange (you need an
organization that provides Exchange accounts to do this, you cannot
set one up yourself).
(Obviously, if you add one of these 3 accounts, you need to add your reminders to these accounts (and not your iCloud account), but you can do that from within the Reminders App. It's really easy to move things between Apple accounts and these accounts, so in minutes you can have everything working again).